= Solution
{c}
Let $n_1,\ldots,n_t$ generate the <submodule> $N$, and let $m_1+N,\ldots,m_s+N$ generate the <quotient module> $M/N$. Choose each $m_i\in M$ as a representative of its class. For any $m\in M$, there are $r_1,\ldots,r_s\in R$ such that
$$
m+N=\sum_{i=1}^s r_i(m_i+N).
$$
Therefore $m-\sum_i r_im_i\in N$. Expressing that difference using the <generating set of a module> $n_1,\ldots,n_t$ gives
$$
m=\sum_{i=1}^s r_im_i+\sum_{j=1}^t u_jn_j.
$$
Hence
$$
\boxed{M=Rm_1+\cdots+Rm_s+Rn_1+\cdots+Rn_t}
$$
is a <finitely generated module>. This proves that <finite generation is preserved by module extensions>.
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